1549901369-Elements_of_Real_Analysis__Denlinger_

2.5 Monotone Sequences 99

It is reasonable to take x 1 = 2 as our first guess. Using a calculator, we
find

n Xn x2 n

1 2 4

( 5)

Xn+-

2 2.25 5.0625 Xn+l =^2 Xn

3 2.2361111... 5.00019290 ...

4 2.2360679 ... 5.000000002 ...

In Part (b) of the proof of Theorem 2.5.11 we showed that {xn} is monotone
decreasing, and this table shows that x~ is already < 5 + 10-^6 , so we can be
sure that, as a decimal approximation to JS, x 4 = 2.2360679 · · · is correct to
six decimal places. Rounding off to six decimal places, we have J5 = x 5
2.236068. 0

*sup AND INF AS LIMITS OF MONOTONE SEQUENCES

Theorem 2.5.13 Let A be a nonempty set of real numbers. Then inf A and
sup A (when they exist) are limits of monotone sequences of elements of A.
More specifically,

(a) !ju= inf A , then 3 monotone decreasing sequence {an} of elements of

A such that an __, u. Moreover, if inf A (j. A then 3 strictly decreasing

sequence {an } of elements of A such that an __, u.

(b) !ju= sup A, then 3 monotone increasing sequence {an} of elements

of A such that an__, u. Moreover, if sup A (j. A then 3 strictly increasing

sequence {an} of elements of A such that an __, u.

*Proof. Suppose A is a nonempty set of real numbers.

(a) Suppose u =inf A.

(i) Case 1 (u EA): In this case, merely take {an} to be the constant

sequence an = u.

(ii) Case 2 (u rf. A): Then \fa EA, u <a. We define the sequence {an}

by mathematical induction, as follows. Since u + 1 is not a lower

bound for A, 3 a 1 E A 3 u < a 1 < u + l. Similarly, neither a1 nor

u +~is a lower bound for A, so 3a 2 EA 3

u < a2 < min { a1, u + ~}.

Now, suppose n EN and an has been defined so that

u < an < min {an-l, u + ~ }.